To determine which expression gives the distance between the points [tex]\((5,1)\)[/tex] and [tex]\((9,-6)\)[/tex], we need to use the distance formula. The distance formula is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
where [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] are the coordinates of the two points.
For the points [tex]\((5, 1)\)[/tex] and [tex]\((9, -6)\)[/tex]:
1. [tex]\(x_1 = 5\)[/tex]
2. [tex]\(y_1 = 1\)[/tex]
3. [tex]\(x_2 = 9\)[/tex]
4. [tex]\(y_2 = -6\)[/tex]
Let's calculate the squared differences for both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] coordinates:
[tex]\[ (x_2 - x_1)^2 = (9 - 5)^2 = 4^2 = 16 \][/tex]
[tex]\[ (y_2 - y_1)^2 = (-6 - 1)^2 = (-7)^2 = 49 \][/tex]
Therefore, the expression to find the distance is:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{16 + 49} = \sqrt{65} \][/tex]
Thus, the corresponding options translate to:
A. [tex]\((5-9)^2+(1+6)^2 \Rightarrow \)[/tex]
[tex]\[ (5-9)^2 \, \text{and} \, (1+6)^2 \][/tex]
B. [tex]\(\sqrt{(5-9)^2+(1-6)^2} \Rightarrow \sqrt{ \, (5-9)^2 \, \text{and} \, (1-6)^2} \]
C. \(\sqrt{(5-9)^2+(1+6)^2} \Rightarrow \sqrt{(5-9)^2 \, \text{and} \, (1+6)^2}\)[/tex]
D. [tex]\((5-9)^2+(1-6)^2 \Rightarrow (5-9)^2 \, \text{and} \, (1-6)^2\)[/tex]
Since
[tex]\[ (5 - 9)^2 = 16 \][/tex]
[tex]\[ (1 + 6)^2 = 49 \][/tex]
The correct expression that represents the distance formula is Option C:
[tex]\[ \sqrt{(5-9)^2+(1+6)^2} \][/tex]
So, the correct answer is:
C. [tex]\(\sqrt{(5-9)^2+(1+6)^2}\)[/tex]
